arXiv subjects

Fengquan Li

Publications and source records attributed to Fengquan Li.

17 recordsLinked to original sources

arXiv2022

The symmetry for two class of steady stratified periodic water waves

In this paper, we mainly consider two class of travelling stratified periodic water waves, one with negative (or without) surface tension and the other with constant Bernoulli's function and stagnation points. We first establish the symmetry result for stratified water waves with negative (or without) surface tension, but without stagnation by using the modified maximum principle. Furthermore, the symmetry property of stratified water waves with constant Bernoulli's function and stagnation points is also obtained provided the monotonic property is known.

math.AP↗

arXiv2021

The singularities for a periodic transport equation

In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+κΛ^αu=0,\quad (t,x)\in R^{+}\times S, $$ where $κ\geq0$, $0<α\leq1$ and $S=[-π,π]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $κ=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $κΛ^αu$, we also prove that the finite time blowup would occur.

math.AP↗

arXiv2021

Uniqueness and stability of steady-state solution with finite energy to the fractal Burgers equation

The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{\fracα{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers' solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state solution, will return to the steady state as $t\rightarrow\infty$.

math.AP↗

arXiv2021

Continuity properties of the data-to-solution map and ill-posedness for a two-component Fornberg-Whitham system

This work studies a two-component Fornberg-Whitham (FW) system, which can be considered as a model for the propagation of shallow water waves. It's known that its solutions depend continuously on their initial data from the local well-posedness result. In this paper, we further show that such dependence is not uniformly continuous in $H^{s}(R)\times H^{s-1}(R)$ for $s>\frac{3}{2}$, but Höler continuous in a weaker topology. Besides, we also establish that the FW system is ill-posed in the critical Sobolev space $H^{\frac{3}{2}}(R)\times H^{\frac{1}{2}}(R)$ by proving the norm inflation.

math.AP↗

arXiv2021

A two-species competition model with mixed dispersal and free boundaries in time-periodic environment

This paper is concerned with a Lotka-Volterra type competition model with free boundaries in time-periodic environment. One species is assumed to adopt nonlocal dispersal and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We show that this free boundary problem with more general growth functions admits a unique solution defined for all time. A spreading-vanishing dichotomy is obtained and criteria for spreading and vanishing are provided. Moreover, under the weak competition condition we provide the long-time asymptotic behavior of solution when spreading occurs.

math.AP↗

arXiv2020

The existence and decay of solitary waves for the Fornberg-Whitham equation

In this paper, we consider the Fornberg-Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in our proof. Besides, we also prove that the family of solitary solutions is orbital stable and decay exponentially when speed wave c is bigger than 1.

math.AP↗

arXiv2020

On symmetry and recovery of steady continuous stratified periodic water waves

This paper considers two-dimensional steady continuous stratified periodic water waves. Firstly, we prove that each streamline must be symmetric about the crest line when it is strictly monotonous between troughs and crests by exploiting the maximum principle and analysis of surface profile. Then, standard Schauder estimates are exploited on the uniform oblique derivative problems to show that all streamlines are real analytic (including the free surface). Based on above symmetry and regularity of streamlines, finally we provide an analytic expansion method to recover the water waves from horizontal velocity on the axis of symmetry and wave height. Most notably, all of results here are suitable not only for small amplitude but also for large amplitude.

math.AP↗

arXiv2020

The well-posedness, blow-up and travelling waves for a two-component Fornberg-Whitham system

In this paper, the two-component Fornberg-Whitham system is studied. We firstly investigate the well-posedness in classical Sobolev Space and show a blow-up scenario by local-in-time a priori estimates, then we present some sufficient conditions on the initial data to lead to wave breaking. Furthermore, we establish analytically the existence of periodic travelling waves.

math.AP↗

arXiv2019

On properties of vertical velocity for 2-D steady water waves

In this article, we mainly investigate the properties of vertical velocity v for two dimensional steady water waves over a flat bed. Firstly we prove the existence of the inflection point for each streamline, then we find the behavior of v along each streamline depends strictly on concavity and convexity of streamline, which contributes to complete Constantin conjecture on v in Stokes wave. And the location of maximum vertical fluid velocity is also proven to be at the inflection point. Besides, we also extend our results to the cases with monotonous vorticity.

math.AP↗

arXiv2018

Well-posedness and peakons for a higher-order $μ$-Camassa-Holm equation

In this paper, we study the Cauchy problem of a higher-order $μ$-Camassa-Holm equation. By employing the Green's function of $(μ-\partial_{x}^{2})^{-2}$, we obtain the explicit formula of the inverse function $(μ-\partial_{x}^{2})^{-2}w$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we prove the existence of global strong solutions and weak solutions. Moreover, we show that the data-to-solution map is Hölder continuous in $H^{s}(\mathbb{S})$, $s\geq 4$, equipped with the $H^{r}(\mathbb{S})$-topology for $0\leq r<s$. Finally, the equation is shown to admit single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

math-ph↗

arXiv2018

Continuity properties of the data-to-solution map for the two-component higher order Camassa-Holm system

This work studies the Cauchy problem of a two-component higher order Camassa-Holm system, which is well-posed in Sobolev spaces $H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$, $s>\frac{7}{2}$ and its solution map is continuous. We show that the solution map is Hölder continuous in $H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$ equipped with the $H^{r}(\mathbb{R})\times H^{r-2}(\mathbb{R})$-topology for $1\leq r<s$, and the Hölder exponent is expressed in terms of $s$ and $r$.

math.AP↗

arXiv2017

On the Cauchy problem for a higher-order $μ$-Camassa-Holm equation

In this paper, we study the Cauchy problem of a higher-order $μ$-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s\geq 4$. Finally, we prove that the equation admits single peakon solutions.

math-ph↗

arXiv2016

A Pohozaev Identity for the Fractional H$\acute{e}$non System

In this paper, we study the Pohozaev identity associated with a H$\acute{e}$non-Lane-Emden system involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^su=|x|^av^p,&x\inΩ, (-\triangle)^sv=|x|^bu^q,&x\inΩ, u=v=0,&x\in R^n\backslashΩ, \end{array} \right. \end{equation} in a star-shaped and bounded domain $Ω$ for $s\in(0,1)$. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases.

math.AP↗

arXiv2015

A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment

In this paper, we study the dynamics of a two-species competition model with two different free boundaries in heterogeneous time-periodic environment, where the two species adopt a combination of random movement and advection upward or downward along the resource gradient. We show that the dynamics of this model can be classified into four cases, which forms a spreading-vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Rough estimates of the asymptotic spreading speed of free boundaries and the long time behavior of solutions are also established when spreading occurs. Furthermore, some sufficient conditions for spreading and vanishing are provided.

math.AP↗

arXiv2015

The diffusive competition problem with a free boundary in heterogeneous time-periodic environment

In this paper, we consider the diffusive competition problem with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, consisting of an invasive species with density $u$ and a native species with density $v$. We assume that $v$ undergoes diffusion and growth in $R^{N}$ , and $u$ exists initially in a ball $B_{h_0}(0)$, but invades into the environment with spreading front $\{r = h(t)\}$. The effect of the dispersal rate $d_1$, the initial occupying habitat $h_0$, the initial density $u_0$ of invasive species $u$, and the parameter $μ$ (see (1.3)) on the dynamics of this free boundary problem are studied. A spreading-vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given.

math.AP↗

arXiv2015

The diffusive competition problem with a free boundary in strong heterogeneous environment and weak heterogeneous environment

In this paper, we consider the diffusive competition problem consisting of an invasive species with density $u$ and a native species with density $v$. We assume that $v$ undergoes diffusion and growth in $[0, \infty)$, and $u$ exists initially in $[0, h_0)$, but invades into the environment with spreading front ${x=h(t)}$. To understand the effect of the dispersal rate $d_1$, the initial occupying habitat $h_0$, the initial density $u_{0}(x)$ of invasive species $(u)$, and the parameter $μ$ (the ratio of the invasion speed of the free boundary and the invasive species gradient at the expanding front) on the dynamics of this free boundary problem, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. A spreading-vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing is provided both in the strong heterogenous environment and weak heterogenous environment. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given.

math.AP↗

arXiv2014

A diffusive logistic problem with a free boundary in time-periodic environment: favorable habitat or unfavorable habitat

We study the diffusive logistic equation with a free boundary in timeperiodic environment. To understand the effect of the dispersal rate $d$, the original habitat radius $h_0$, the spreading capability $μ$, and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d$, $h_0$, $μ$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $λ_1(d, α, γ, h(t), T)$ to determine the spreading and vanishing of the invasive species. We prove that if $λ_1(d, α, γ, h_0, T)\leq 0$, the spreading must happen; while if $λ_1(d, α, γ, h_0, T)> 0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the dispersal rate is slow or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small, while survive successfully if the initial value is big. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.

math.AP↗